Invasive species cause significant losses through their effect on agriculture, human health and the environment. Their importance has increased with time owing to globalisation, as the spread of invasive species is facilitated by the increased movement of people, cargo and genetic material around the world. There is a vast literature on the economics of invasive species and their management. Here, we contribute to this literature by applying a spatio-temporal model to the allocation of surveillance resources. We focus on three questions regarding resource allocation to control a newly discovered invasion: the budget, which determines the amount of search effort available; the duration of the control program; and the allocation of surveillance and control in time and space. We also explore the complementary role of passive surveillance by members of the public. We derive efficient frontiers for effort allocation that represent the trade-off between cost and probability of eradication after inefficient strategies have been eliminated. We use the results to illustrate how to evaluate whether introduction of passive surveillance is desirable based on cost and eradication probability. We conclude by discussing the implications of our findings in the design of control programs.